[Bridges] use dual of equality in slack bridge (take 2)

src/Bridges/Constraint/bridges/slack.jl

@@ -124,12 +124,25 @@
 function MOI.get(
     model::MOI.ModelLike,
     a::Union{MOI.ConstraintDual,MOI.ConstraintDualStart},
-    bridge::_AbstractSlackBridge,
-)
+    bridge::_AbstractSlackBridge{T},
+) where {T}
     # The dual constraint on slack (since it is free) is
-    # -dual_slack_in_set + dual_equality = 0 so the two duals are
+    # `-dual_slack_in_set + dual_equality = 0` so the two duals are
     # equal and we can return either one of them.
-    return MOI.get(model, a, bridge.slack_in_set)
+    # We decide to use the dual of the equality constraints because
+    # the `slack_in_set` constraint might be bridge by a variable bridge that
+    # does not # support `ConstraintDual`. This is the case if the adjoint of
+    # the linear map on which the bridge is based is not invertible, as for
+    # instance with `Variable.ZerosBridge` or
+    # `SumOfSquares.Bridges.Variable.KernelBridge`.
+    dual = MOI.get(model, a, bridge.equality)
+    if dual === nothing
+        return nothing
+    elseif dual isa AbstractVector
+        scale = MOI.Utilities.SetDotScalingVector{T}(bridge.set)
+        return dual ./ scale.^2
+    end
+    return dual
 end
 
 function MOI.set(
@@ -209,6 +222,7 @@
 struct ScalarSlackBridge{T,F,S} <:
        _AbstractSlackBridge{T,MOI.VariableIndex,MOI.EqualTo{T},F,S}
     slack::MOI.VariableIndex
+    set::S
     slack_in_set::MOI.ConstraintIndex{MOI.VariableIndex,S}
     equality::MOI.ConstraintIndex{F,MOI.EqualTo{T}}
 end
@@ -226,7 +240,7 @@
     slack, slack_in_set = MOI.add_constrained_variable(model, s)
     new_f = MOI.Utilities.operate(-, T, f, slack)
     equality = MOI.add_constraint(model, new_f, MOI.EqualTo(zero(T)))
-    return ScalarSlackBridge{T,F,S}(slack, slack_in_set, equality)
+    return ScalarSlackBridge{T,F,S}(slack, s, slack_in_set, equality)
 end
 
 # Start by allowing all scalar constraints:
@@ -341,6 +355,7 @@
 struct VectorSlackBridge{T,F,S} <:
        _AbstractSlackBridge{T,MOI.VectorOfVariables,MOI.Zeros,F,S}
     slack::Vector{MOI.VariableIndex}
+    set::S
     slack_in_set::MOI.ConstraintIndex{MOI.VectorOfVariables,S}
     equality::MOI.ConstraintIndex{F,MOI.Zeros}
 end
@@ -358,7 +373,7 @@
     slack, slack_in_set = MOI.add_constrained_variables(model, s)
     new_f = MOI.Utilities.operate(-, T, f, MOI.VectorOfVariables(slack))
     equality = MOI.add_constraint(model, new_f, MOI.Zeros(d))
-    return VectorSlackBridge{T,F,S}(slack, slack_in_set, equality)
+    return VectorSlackBridge{T,F,S}(slack, s, slack_in_set, equality)
 end
 
 function MOI.supports_constraint(